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Local Connectivity of the Mandelbrot Set at Certain Infinitely Renormalizable Points∗†

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Local Connectivity of the Mandelbrot Set at Certain Infinitely Renormalizable Points∗† Local connectivity of the Mandelbrot set at certain infinitely renormalizable points¤y Yunping Jingz Abstract We construct a subset consisting of infinitely renormalizable points in the Mandelbrot set. We show that Mandelbrot set is locally connected at this subset and for every point in this sub- set, corresponding infinitely renormalizable quadratic Julia set is locally connected. Since the set of Misiurewicz points is in the closure of the subset we construct, therefore, the subset is dense in the boundary of the Mandelbrot set. Contents 1 Introduction, review, and statements of new results 237 1.1 Quadratic polynomials . 237 1.2 Local connectivity for quadratic Julia sets. 240 1.3 Some basic facts about the Mandelbrot set from Douady- Hubbard . 244 1.4 Constructions of three-dimensional partitions. 248 1.5 Statements of new results. 250 2 Basic idea in our construction 252 3 The proof of our main theorem 259 ¤2000 Mathematics Subject Classification. Primary 37F25 and 37F45, Secondary 37E20 yKeywords: infinitely renormalizable point, local connectivity, Yoccoz puzzle, the Mandelbrot set zPartially supported by NSF grants and PSC-CUNY awards and 100 men program of China 236 Complex Dynamics and Related Topics 237 1 Introduction, review, and statements of new results 1.1 Quadratic polynomials Let C and C be the complex plane and the extended complex plane (the Riemann sphere). Suppose f is a holomorphic map from a domain Ω ½ C into itself. A point z in Ω is called a periodic point of period k ¸ 1 if f ±k(z) = z but f ±i(z) 6= z for all 1 · i < k. The number ¸ = (f ±k)0(z) is called the multiplier of f at z. A periodic point of period 1 is also called a fixed point. A periodic point z of f is said to be super-attracting, attracting, neutral, or repelling if j¸j = 0, 0 < j¸j < 1, j¸j = 1, or j¸j > 1. The proof of the following theorem can be found in Milnor’s book [22, pp. 86-88]. Theorem 1.1 (Theorem of B¨ottcher). Suppose p is a super-attracting periodic point of period k of a holomorphic map f from a domain Ω into itself. There is a neighborhood W of p, a holomorphic diffeomorphism h : W ! h(W ) with h(p) = 0, and a unique integer n > 1 such that h ± f ±k ± h¡1(w) = wn for w 2 h(W ). Furthermore, h is unique up to multiplication by an (n ¡ 1)-st root of unity. Consider a quadratic polynomial f(z) = az2 + bz + d. Conjugating f by an appropriate linear map h(z) = ez +s, we get h±f ±h¡1(z) = z2 +c. So from dynamical systems point of view, quadratic polynomials form a 2 one-parameter family. We call qc(z) = z +c the Douady-Hubbard family of quadratic polynomials. When we study this family, we always deal with two kinds of sets, one is the class of sets in the phase space (the z-plane) and the other is the class of sets in the parameter space (the c-plane). In this paper, we use regular letters to denote sets in the phase space and curly letters to denote sets in the parameter space. Denote Dr = fz 2 C j jzj < rg the disk of radius r centered 0. Let 2 ¡1 qc(z) = z +c be a quadratic polynomial. For r large enough, U = qc (Dr) is a simply connected domain and its closure is relatively compact in Dr. Then qc : U ! V = Dr is a holomorphic, proper, degree two branched covering map. This is a model of quadratic-like maps defined by Douady and Hubbard [2] as follows: A quadratic-like map is a triple (f; U; V ) such that U and V are simply connected domains isomorphic to a disc Complex Dynamics and Related Topics 238 with U ½ V and such that f : U ! V is a holomorphic, proper, degree two branched covering mapping. For a quadratic-like map (f; U; V ), 1 ¡n Kf = \n=0f (U) is called the filled-in Julia set. The Julia set Jf is the boundary of Kf . Both Kf and Jf are compact. A quadratic-like map (f; U; V ) has only one critical point which we always denote as 0. Refer to [22, pp. 91-92] for a proof of the following theorem. Theorem 1.2. The set Kf (as well as Jf ) is connected if and only if 0 is in Kf . Moreover, if 0 is not in Kf , Kf = Jf is a Cantor set. Since qc : U ! V = Dr is a quadratic-like map, the filled-in Julia set of qc is the set of all points not going to infinity under forward iterates of qc. We use Kc and Jc to mean its filled-in Julia set and Julia set. For c = 0 and every r > 1, (q0;Dr;Dr2 ) is a quadratic-like map whose filled- in Julia set is K0 = D1. For any c, 1 is a super-attracting fixed point of qc, applying Theorem 1.1, there is a unique holomorphic diffeomorphism hc (called the B¨ottcher coordinate) defined on a neighborhood B0 about 0 1 with hc(1) = 1, hc(1) = 1 such that ¡1 2 hc ± qc ± hc (z) = z ; z 2 B0: Assume hc(B0) = C n Dr (for a fixed large number r > 1). Let Bn = ¡n qc (B0). If 0 62 Bn, then qc : Bn \ C ! Bn¡1 \ C; n ¸ 1; is unramified covering map of degree two. So we can inductively extend hc : Bn ! C n D 1 r 2n such that ¡1 2 hc ± qc ± hc (z) = z z 2 Bn: Let ±n Bc(1) = fz 2 C j qc (z) ! 1 as n ! 1g be the basin of 1. Then Kc = C n Bc(1). Let + ±n log jqc (z)j Gc(z) = lim ; n!1 2n Complex Dynamics and Related Topics 239 + where log x = supf0; log xg, be the Green function of Kc. It is a proper harmonic function whose zero set is Kc and whose critical points are bounded by Gc(0). So the B¨ottcher coordinate can be extended to hc : Uc = fz 2 C j Gc(z) > Gc(0)g ! fz 2 C j jzj > exp Gc(0)g: such that ¡1 2 hc ± qc ± hc (z) = z : ¡1 Moreover, Gc = log jhcj. The set Gc (log r), r > 1, is called an equipo- tential curve. If Kc is connected (equivalent to say 0 62 Bc(1)), then Uc = Bc(1) n f1g; if Kc is a Cantor set (equivalent to say 0 2 Bc(1)), then Uc is a punctured disk bounded by an 1 shape curve passing 0. By the definition, the Mandelbrot set M is defined as the set of parameters c such that Kc is connected. Suppose c 2 M. Let Sr = fz 2 C j jzj = rg be a circle of radius r > 1. Then ¡1 ¡1 sr = hc (Sr) = Gc (log r) is an equipotential curve. Note that qc(sr) = sr2 : ¡1 For every r > 1, let Ur = hc (Dr). Then Ur is a domain bounded by the equipotential curve sr. Furthermore, (qc;Ur;Ur2 ) is a quadratic-like map whose filled-in Julia set is Kc. In the rest of this paper, when we talk about a quadratic polynomial q(z) = z2 + c, it also mean a quadratic-like map (q; Ur;Ur2 ) for any r > 1. Take Eθ = fz 2 C j jzj > 1; arg(z) = 2¼θg for 0 · θ < 1. Let ¡1 eθ = hc (Eθ): Then eθ is called an external ray of angle θ and qc(eθ) = e2θ (mod 1): An external ray is an integral curve of the gradient vector field rG on Bc(1). An external ray eθ is said to land at Jc if eθ has only one limiting ±i point at Jc. It is periodic with period m if qc (eθ) \ eθ = ; for 1 · i < m ±m and if qc (eθ) = eθ. Refer to [9, 23] for the following theorem (the reader can also find a proof in [10, pp. 199]. Complex Dynamics and Related Topics 240 2 Theorem 1.3 (Douady and Yoccoz). Suppose qc(z) = z + c is a quadratic polynomial with a connected filled-in Julia set Kc. Every repelling periodic point of qc is a landing point of finitely many periodic external rays with the same period. Two quadratic-like maps (f; U; V ) and (g; U 0;V 0) are said to be topo- logically conjugate if there is a homeomorphism h from a neighborhood 0 Kf ½ X ½ U to a neighborhood Kg ½ Y ½ U such that h ± f(z) = g ± h(z); z 2 X: If h is quasiconformal (see [1]) (resp. holomorphic), then they are quasi- conformally (resp. holomorphically) conjugate. If h can be chosen such that hz = 0 a.e. on Kf , then they are hybrid equivalent. The reader can find the proof of the following theorem in [2]. Theorem 1.4 (Douady and Hubbard). Suppose (f; U; V ) is a quadratic- like map whose Julia set Jf is connected. Then there is a unique quadratic 2 polynomial qc(z) = z + c which is hybrid equivalent to f. The following basic facts about the Julia set of a quadratic-like map (f; U; V ) will be used in the paper but the reader can check them by himself by referring to [22]. The Julia set Jf is completely invariant, ¡1 i.e., f(Jf ) = Jf and f (Jf ) = Jf .
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